# Math Help - Gaussian Integral

1. ## Gaussian Integral

Heya having a complete mind blank today can anyone give me a hint on how to get started with this
$
\int_0^\infty x^2e^{-x^2}dx
$

Cheers

2. Let $u=x^{2}, \;\ \frac{1}{2}du=xdx$

Make the subs and we get a form we can easily use the gamma function.

We get upon making the subs:

$\frac{1}{2}\int_{0}^{\infty}u^{\frac{1}{2}}e^{-u}du$

Now, use ${\Gamma}(p)=\int_{0}^{\infty}u^{p-1}e^{-u}du$

See what p must be?. p-1=1/2

3. Hello,

$\int_0^\infty x^2 e^{-x^2} ~ dx=-\frac 12 \int x(-2x)e^{-x^2} ~ dx$
Integrate by parts :
$=-\frac 12 \left[xe^{-x^2}\right]_0^\infty+\frac 12 \int_0^\infty e^{-x^2} ~ dx=\frac 12 \int_0^\infty e^{-x^2} ~ dx$

Knowing that $\int_0^\infty e^{-x^2} ~ dx=\frac{\sqrt{\pi}}{2}$, we then have :

$I=\frac{\sqrt{\pi}}{4}$